nLab artinian ring

Contents

Contents

Definition

Artinian rings

Every ring RR has a canonical RR-RR-bimodule structure, with left action α L:R×RR\alpha_L:R \times R \to R and right action α R:R×RR\alpha_R:R \times R \to R defined as the multiplicative binary operation on RR and biaction α:R×R×RR\alpha:R \times R \times R \to R defined as the ternary product on RR:

α L(a,b)ab\alpha_L(a, b) \coloneqq a \cdot b
α R(a,b)ab\alpha_R(a, b) \coloneqq a \cdot b
α(a,b,c)abc\alpha(a, b, c) \coloneqq a \cdot b \cdot c

Let TwoSidedIdeals(R)\mathrm{TwoSidedIdeals}(R) be the category of two-sided ideals in RR, whose objects are two-sided ideals II in RR, sub- R R - R R -bimodules of RR with respect to the canonical bimodule structure on RR, and whose morphisms are RR-RR-bimodule monomorphisms.

A descending chain of two-sided ideals in RR is an inverse sequence of two-sided ideals in RR, a sequence of two-sided ideals A:TwoSidedIdeals(R)A:\mathbb{N} \to \mathrm{TwoSidedIdeals}(R) with the following dependent sequence of RR-RR-bimodule monomorphisms: for natural number nn \in \mathbb{N}, a dependent RR-RR-bimodule monomorphism i n:A n+1A ni_n:A_{n+1} \hookrightarrow A_{n}.

A ring RR is Artinian if it satisfies the descending chain condition on its two-sided ideals: for every descending chain of two-sided ideals (A,i n)(A, i_n) in RR, there exists a natural number mm \in \mathbb{N} such that for all natural numbers nmn \geq m, the RR-RR-bimodule monomorphism i n:A n+1A ni_n:A_{n+1} \hookrightarrow A_{n} is an RR-RR-bimodule isomorphism.

Left Artinian rings

Let LeftIdeals(R)\mathrm{LeftIdeals}(R) be the category of left ideals in RR, whose objects are left ideals II in RR, sub-left-RR-modules of RR with respect to the canonical left module structure ()():R×RR(-)\cdot(-):R \times R \to R on RR, and whose morphisms are left RR-module monomorphisms.

A descending chain of left ideals in RR is an inverse sequence of left ideals in RR, a sequence of left ideals A:LeftIdeals(R)A:\mathbb{N} \to \mathrm{LeftIdeals}(R) with the following dependent sequence of left RR-module monomorphisms: for natural number nn \in \mathbb{N}, a dependent left RR-module monomorphism i n:A n+1A ni_n:A_{n+1} \hookrightarrow A_{n}.

A ring RR is left Artinian if it satisfies the descending chain condition on its left ideals: for every descending chain of left ideals (A,i n)(A, i_n) in RR, there exists a natural number mm \in \mathbb{N} such that for all natural numbers nmn \geq m, the left RR-module monomorphism i n:A n+1A ni_n:A_{n+1} \hookrightarrow A_{n} is an left RR-module isomorphism.

Right Artinian rings

Let RightIdeals(R)\mathrm{RightIdeals}(R) be the category of right ideals in RR, whose objects are right ideals II in RR, sub-right-RR-modules of RR with respect to the canonical right module structure ()():R×RR(-)\cdot(-):R \times R \to R on RR, and whose morphisms are right RR-module monomorphisms.

A descending chain of right ideals in RR is an inverse sequence of right ideals in RR, a sequence of right ideals A:RightIdeals(R)A:\mathbb{N} \to \mathrm{RightIdeals}(R) with the following dependent sequence of right RR-module monomorphisms: for natural number nn \in \mathbb{N}, a dependent right RR-module monomorphism i n:A n+1A ni_n:A_{n+1} \hookrightarrow A_{n}.

A ring RR is right Artinian if it satisfies the descending chain condition on its right ideals: for every descending chain of right ideals (A,i n)(A, i_n) in RR, there exists a natural number mm \in \mathbb{N} such that for all natural numbers nmn \geq m, the right RR-module monomorphism i n:A n+1A ni_n:A_{n+1} \hookrightarrow A_{n} is an right RR-module isomorphism.

Properties

In an artinian ring RR the Jacobson radical J(R)J(R) is nilpotent. A left artinian ring is semiprimitive if and only if the zero ideal is the unique nilpotent ideal.

Artinian and Noetherian rings

A dual condition is noetherian: a noetherian ring is a ring satisfying the ascending chain condition on ideals. The symmetry is severely broken if one considers unital rings: for example every unital artinian ring is noetherian. For a converse there is a strong condition: a left (unital) ring RR is left artinian iff R/J(R)R/J(R) is semisimple in RMod_R Mod and the Jacobson radical J(R)J(R) is nilpotent. Artinian rings are intuitively much smaller than generic noetherian rings.

See also

commutative ringreduced ringintegral domain
local ringreduced local ringlocal integral domain
Artinian ringsemisimple ringfield
Weil ringfieldfield

References

Last revised on January 12, 2023 at 17:08:11. See the history of this page for a list of all contributions to it.